A084357 -id:A084357 - cp's OEIS frontend

A122949 Number of ordered pairs of permutations generating a transitive group.

1, 3, 26, 426, 11064, 413640, 20946960, 1377648720, 114078384000, 11611761920640, 1425189271161600, 207609729886944000, 35419018603306060800, 6996657393055480550400, 1584616114318716544665600, 407930516160959891683584000, 118458533875304716189544448000

Offset: 1

Philippe Flajolet, Oct 25 2006

nonn

From Dixon: The sequence is asymptotic to (n!)^2; when divided by n!^2, it has a high-order asymptotic contact with the probability that two randomly chosen permutations generate the symmetric group. Also: a(n)=(n-1)!*A003319(n+1), where A003319 is the number of connected [or indecomposable] permutations. The coefficients in the asymptotic expansion of a(n)/(n!)^2 are A113869 and in absolute value, they constitute A084357 (number of sets of sets of lists).

			a(2)=3 because there are 2!*2!=4 pairs of permutations, of which only [(1,1),(1,1)] does not generate a transitive group.

Seiichi Manyama, Table of n, a(n) for n = 1..253
John D. Dixon, Asymptotics of Generating the Symmetric and Alternating Groups, Electronic Journal of Combinatorics, vol 11(2), R56.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; page 139.

Cf. A003319, A084357, A113869.

series(log(add(n!*z^n,n=0..Order+2)),z=0):seq(coeff(%,z,j)*j!,j=0..Order);

max = 15; Drop[ CoefficientList[ Series[ Log[1 + Sum[n!*z^n, {n, 1, max}]], {z, 0, max}], z]* Range[0, max]!, 1](* Jean-François Alcover, Oct 05 2011 *)

N=20; x='x+O('x^N); Vec(serlaplace(log(sum(k=0, N, k!*x^k)))) \\ Seiichi Manyama, Mar 01 2019

Exponential generating function is: log(1+Sum_{n>=1}n!*z^n).

a(n) = (n!)^2 - (n-1)! * Sum_{k=1..n-1} a(k) * (n-k)! / (k-1)!. - Ilya Gutkovskiy, Jul 10 2020

More terms from Seiichi Manyama, Mar 01 2019

A075744 Hierarchies of hierarchies.

Original entry on oeis.org

1, 1, 5, 36, 338, 3898, 53173, 835992, 14864340, 294606273, 6434871231, 153473830678, 3966604562709, 110386840008838, 3289768253831145, 104502173165838799, 3523895395660532937, 125689588963370029163, 4726867751402704638366, 186902021178952943036080

Offset: 0

N. J. A. Sloane, Oct 15 2002

nonn

Stirling transform of A084357 = number of sets of sets of lists. - Thomas Wieder, Jun 19 2005

T. D. Noe, Table of n, a(n) for n=0..100

Cf. A075729, A075756.

Cf. A084357.

m = 20;
f[x_] = Exp[1/(2 - Exp[x]) - 1];
CoefficientList[Exp[f[x] - 1] + O[x]^m, x]*Range[0, m - 1]! (* Jean-François Alcover, Feb 24 2019 *)

E.g.f.: exp(f(x)-1) where f(x) = e.g.f. for A075729.

A088814 Matrix product of unsigned Lah-triangle |A008297(n,k)| and Stirling2-triangle A008277(n,k).

Original entry on oeis.org

1, 3, 1, 13, 9, 1, 73, 79, 18, 1, 501, 755, 265, 30, 1, 4051, 7981, 3840, 665, 45, 1, 37633, 93135, 57631, 13580, 1400, 63, 1, 394353, 1192591, 911582, 274141, 38290, 2618, 84, 1, 4596553, 16645431, 15285313, 5633922, 999831, 92358, 4494, 108, 1, 58941091

Offset: 1

Vladeta Jovovic, Nov 22 2003

Also the Bell transform of A000262(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

Cf. A000262(first column), A084357(row sums).

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> simplify(hypergeom([-n,-n-1],[],1)), 9); # Peter Luschny, Jan 26 2016

BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
B = BellMatrix[Function[n, Sum[BellY[n+1, k, Range[n+1]!], {k, 0, n+1}]], rows];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny_ *)

E.g.f.: exp(y*(exp(x/(1-x))-1)).

A317362 Expansion of e.g.f. exp(exp(x/(1 + x)) - 1).

Original entry on oeis.org

1, 1, 0, -1, 3, -8, 23, -89, 556, -4773, 44425, -397670, 3060577, -12448655, -235761640, 9571505555, -241952653453, 5424619822460, -116900288145113, 2494797839905055, -53406941947725348, 1152770311462756071, -25109138533156554399, 550613923917090815374, -12088287036694435407999

Offset: 0

Ilya Gutkovskiy, Jul 26 2018

sign

Inverse Lah transform of the Bell numbers (A000110).

N. J. A. Sloane, Transforms

Cf. A000110, A084357.

a:= proc(n) option remember; add((-1)^(n-k)*n!/k!*
      binomial(n-1, k-1)*combinat[bell](k), k=0..n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 26 2018

nmax = 24; CoefficientList[Series[Exp[Exp[x/(1 + x)] - 1], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^(n - k) Binomial[n - 1, k - 1] BellB[k] n!/k!, {k, 0, n}], {n, 0, 24}]

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n-1,k-1)*Bell(k)*n!/k!, where Bell() = A000110.

A317366 Expansion of e.g.f. exp(exp(x/(1 - x)) - 1)/(1 - x).

Original entry on oeis.org

1, 2, 8, 47, 359, 3347, 36665, 460098, 6494444, 101708007, 1748263435, 32697711895, 660642793717, 14332871438810, 332186039584768, 8188070581358795, 213821204277955267, 5895325327054011087, 171095582314380667621, 5212792218964517899506, 166321395872186089502972, 5545223090189205308551443

Offset: 0

Ilya Gutkovskiy, Jul 26 2018

nonn

Cf. A000110, A002720, A084357, A101053.

a:= proc(n) option remember; add(binomial(n, k)^2
      *k!*combinat[bell](n-k), k=0..n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 26 2018

nmax = 21; CoefficientList[Series[Exp[Exp[x/(1 - x)] - 1]/(1 - x) , {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k]^2 k! BellB[n - k], {k, 0, n}], {n, 0, 21}]

a(n) = Sum_{k=0..n} binomial(n,k)^2*k!*Bell(n-k), where Bell() = A000110.

A332024 E.g.f.: Product_{k>=1} (1 + x^k/(k!*(1 - x)^k)).

Original entry on oeis.org

1, 1, 3, 16, 113, 956, 9382, 105253, 1334517, 18904936, 295787126, 5056826039, 93594929738, 1861321879535, 39536014577711, 892763601542509, 21352130132268541, 539243894127067888, 14342761454293102006, 400830115867761118963, 11743833994363640228070

Offset: 0

Ilya Gutkovskiy, Feb 13 2020

nonn

Cf. A007837, A084357, A129519, A320567, A330649.

nmax = 20; CoefficientList[Series[Product[(1 + x^k/(k! (1 - x)^k)), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

seq(n)={Vec(serlaplace(prod(k=1, n, (1 + x^k/(k!*(1 - x)^k)) + O(x*x^n))))} \\ Andrew Howroyd, Feb 13 2020

a(n) = Sum_{k=0..n} binomial(n-1,k-1) * A007837(k) * n! / k!.

A256892 Triangular array read by rows, the matrix product of the unsigned Lah numbers and the Stirling set numbers, T(n,k) for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 13, 9, 1, 0, 73, 79, 18, 1, 0, 501, 755, 265, 30, 1, 0, 4051, 7981, 3840, 665, 45, 1, 0, 37633, 93135, 57631, 13580, 1400, 63, 1, 0, 394353, 1192591, 911582, 274141, 38290, 2618, 84, 1, 0, 4596553, 16645431, 15285313, 5633922, 999831, 92358, 4494, 108, 1

Offset: 0

Peter Luschny, Apr 12 2015

Also the Bell transform of A000262(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 29 2016

			Triangle starts:
1;
0,    1;
0,    3,    1;
0,   13,    9,    1;
0,   73,   79,   18,   1;
0,  501,  755,  265,  30,  1;
0, 4051, 7981, 3840, 665, 45, 1;

See also A088814 and A088729 for variants based on an (1,1)-offset of the number triangles. See A131222 for the product Lah * Stirling-cycle.
A079640 is an unsigned matrix inverse reduced to an (1,1)-offset.
Cf. A000262, A045943, A084357, A088729, A088814.

# The function BellMatrix is defined in A264428.
BellMatrix(n -> simplify(hypergeom([-n, -n-1], [], 1)), 9); # Peter Luschny, Jan 29 2016

BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[Function[n, HypergeometricPFQ[{-n, -n-1}, {}, 1]], rows = 12];
Table[B[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

def Lah(n, k):
    if n == k: return 1
    if k<0 or  k>n: return 0
    return (k*n*gamma(n)^2)/(gamma(k+1)^2*gamma(n-k+1))
matrix(ZZ, 8, Lah) * matrix(ZZ, 8, stirling_number2)  # as a square matrix

T(n+1,1) = A000262(n).
T(n+1,n) = A045943(n).
Row sums are A084357.

A308517 Expansion of e.g.f. exp(1 - exp(x/(1 - x))).

Original entry on oeis.org

1, -1, -2, -5, -11, 18, 711, 10113, 125042, 1485627, 17151083, 185932580, 1665928529, 4570649471, -349942007986, -14532197609433, -433111168649251, -11579368513540914, -293948221716443209, -7208510256850719447, -170577027262193604678, -3823168355141657356481, -76959686241473750407701

Offset: 0

Ilya Gutkovskiy, Jun 03 2019

sign

Lah transform of A000587 (complementary Bell numbers).

Cf. A000587, A084357.

nmax = 22; CoefficientList[Series[Exp[1 - Exp[x/(1 - x)]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] BellB[k, -1] n!/k!, {k, 0, n}]; Table[a[n], {n, 0, 22}]

a(n) = Sum_{k=0..n} binomial(n-1,k-1)*A000587(k)*n!/k!.

cp's OEIS Frontend

A122949 Number of ordered pairs of permutations generating a transitive group.

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A075744 Hierarchies of hierarchies.

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A088814 Matrix product of unsigned Lah-triangle |A008297(n,k)| and Stirling2-triangle A008277(n,k).

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A317362 Expansion of e.g.f. exp(exp(x/(1 + x)) - 1).

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A317366 Expansion of e.g.f. exp(exp(x/(1 - x)) - 1)/(1 - x).

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A332024 E.g.f.: Product_{k>=1} (1 + x^k/(k!*(1 - x)^k)).

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A256892 Triangular array read by rows, the matrix product of the unsigned Lah numbers and the Stirling set numbers, T(n,k) for n>=0 and 0<=k<=n.

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A308517 Expansion of e.g.f. exp(1 - exp(x/(1 - x))).

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